Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits

With a probability of success of 95% we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres–Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be separable, free entangled or bound entangled. Leveraging a geometrical representation of these states in Euclidean space, novel methods are presented that allow the classification of separable and bound entangled Bell diagonal states in an efficient way. Moreover, the classification allows the precise determination of relative volumes of the classes of separable, free and bound entangled states. In detail, out of all Bell diagonal PPT states \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$81.0\% \pm 0.1\%$$\end{document}81.0%±0.1% are determined to be separable while \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$13.9 \pm 0.1\%$$\end{document}13.9±0.1% are bound entangled and only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.1 \pm 0.1\%$$\end{document}5.1±0.1% remain unclassified. Moreover, our applied criteria are compared for their effectiveness and relation as detectors of bound entanglement, which reveals that not a single criterion is capable to detect all bound entangled states.

(1) Here the first relation is the assumption and the second follows from (1). So we also have: Denote the set of bound entangled states by BOUND. A similar argument shows that bound states are again mapped to bound states: By definition of entanglement conserving symmetries we have Now suppose s : Again the first relations is the assumption and the second from (3). Together with (1) this shows that bound entangled states are mapped to entangled states which are also PPT: (2), (3) and (4) together show the invariance of the classes free entangled, separable and bound entangled under the entanglement conserving symmetries.

A2: Parameterization of separable states to determine bounds for numeric EWs
An efficient parameterization of the set of separable states is required for entanglement detection with numeric EWs (E5). Using an EW W requires the determination of its upper and lower bound for the set of separable states ρ s ∈ SEP: L ≤ tr[ρ s W ] ≤ U.
To determine these bounds numerically, a parameterization of separable state is required. Due to the linearity of the trace and the fact that general separable states are defined as convex mixtures of pure states, it suffices to consider pure separable states Any such state can be generated by a local unitary transformation of the state |00 00| ≡ |0 0| ⊗ |0 0|: In Ref. 40 an efficient parameterization of unitaries was proposed that requires only 2(d − 1) parameters to construct any pure state ρ 1/2 . Consequently, defining a separable, bipartite state ρ s requires 4(d − 1) parameters. Naturally, also mixtures of K pure states can be generated accordingly by using 4K(d − 1) parameters to generate K pure states to be mixed with probabilities p 1 , . . . , p K . The optimization over all separable states to find ρ min/max for a given EW W is carried out for the parameters of the composite parameterization of unitaries according to (5). For d = 3, 8 independent parameters need to be optimized in a bounded region. This is done numerically, using the implementation of the "Optim" 43 package of the "LBFGS" algorithm which uses the gradient and an approximation of the Hessian of tr[W ρ]. To minimize the risk of identifying only a local maximum/minimum, the optimization procedures is performed 50 times with random starting points of the algorithm and the overall minimum/maximum is taken for L/U.

A3: Representation for the generators of the considered symmetry group
The details of the construction of (anti-)unitary transformations corresponding to the generators of the considered symmetry groups has been introduced in Ref. 24 . They act as permutations of the Bell basis element P k,l and are applied to any diagonal state in the Bell basis by multiplication of the diagonal elements with the according permutation matrix. Let c be the coordinate vector collecting the elements c k,l , k, l = 0, 1, 2 of the diagonal density matrix of a state in M 3 represented in the Bell basis {P k,l }. The symmetries act as permutation on the basis elements P k,l or equivalently on the elements c k,l with the inverse permutation. For a symmetry s acting on the coordinates c with permutation matrix M s , the coordinatesc of the transformed state are then given byc = M s · c. Enumerating the basis elements as P 0,0 , P 1,0 , P 2,0 , P 2,1 , . . . , P 1,2 , P 2,2 , the permutation matrices for the generators are given below (all indices are defined by (mod 3)).
Momentum inversion: m : P k,l → P −k,l : Quarter rotation: r : P k,l → P k,−l , Vertical sheer: v : P k,l → P k+l,l ,